Question: Simplify and expand the following expression: $ \dfrac{1}{5y - 20}+ \dfrac{4}{5y + 40}+ \dfrac{y}{y^2 + 4y - 32} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the first term: $ \dfrac{1}{5y - 20} = \dfrac{1}{5(y - 4)}$ We can factor a $5$ out of denominator in the second term: $ \dfrac{4}{5y + 40} = \dfrac{4}{5(y + 8)}$ We can factor the quadratic in the third term: $ \dfrac{y}{y^2 + 4y - 32} = \dfrac{y}{(y - 4)(y + 8)}$ Now we have: $ \dfrac{1}{5(y - 4)}+ \dfrac{4}{5(y + 8)}+ \dfrac{y}{(y - 4)(y + 8)} $ The least common multiple of the denominators is: $ 25(y - 4)(y + 8)$ In order to get the first term over $25(y - 4)(y + 8)$ , multiply by $\dfrac{5(y + 8)}{5(y + 8)}$ $ \dfrac{1}{5(y - 4)} \times \dfrac{5(y + 8)}{5(y + 8)} = \dfrac{5(y + 8)}{25(y - 4)(y + 8)} $ In order to get the second term over $25(y - 4)(y + 8)$ , multiply by $\dfrac{5(y - 4)}{5(y - 4)}$ $ \dfrac{4}{5(y + 8)} \times \dfrac{5(y - 4)}{5(y - 4)} = \dfrac{20(y - 4)}{25(y - 4)(y + 8)} $ In order to get the third term over $25(y - 4)(y + 8)$ , multiply by $\dfrac{25}{25}$ $ \dfrac{y}{(y - 4)(y + 8)} \times \dfrac{25}{25} = \dfrac{25y}{25(y - 4)(y + 8)} $ Now we have: $ \dfrac{5(y + 8)}{25(y - 4)(y + 8)} + \dfrac{20(y - 4)}{25(y - 4)(y + 8)} + \dfrac{25y}{25(y - 4)(y + 8)} $ $ = \dfrac{ 5(y + 8) + 20(y - 4) + 25y} {25(y - 4)(y + 8)} $ Expand: $ = \dfrac{5y + 40 + 20y - 80 + 25y}{25y^2 + 100y - 800} $ $ = \dfrac{50y - 40}{25y^2 + 100y - 800}$ Simplify: $ = \dfrac{10y - 8}{5y^2 + 20y - 160}$